A133214 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(6 + 2\,x + 4\,x^{2} + 11\,n + 9\,n x + 2\,n x^{2} + 8\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(18 + 18\,x + 28\,x^{2} + 16\,x^{3} + 25\,n + 65\,n x + 52\,n x^{2} + 12\,n x^{3} + 12\,n^{2} + 52\,n^{2} x + 24\,n^{2} x^{2} + 2\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(12\,x - 14\,x^{2} - 16\,x^{3} - 8\,x^{4} + 14\,n x - 20\,n x^{2} - 16\,n x^{3} + 4\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 9 + 16\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	9	16	4								29
1	19	90	60	8							177
2	33	320	528	192	16						1089
3	51	850	2800	2400	560	32					6693
4	73	1872	10500	17600	9360	1536	64				41005
5	99	3626	31164	88200	90160	32928	4032	128			250337
6	129	6400	78400	338688	580160	401408	107520	10240	256		1523201
7	163	10530	174528	1072512	2794176	3217536	1612800	331776	25344	512	9239877

All 15 rows

Technical details

Node id: n-af0fa7591e93aaa0 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e8b03ce9b132c5b4

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   9,    16,      4;
  1:  19,    90,     60,       8;
  2:  33,   320,    528,     192,       16;
  3:  51,   850,   2800,    2400,      560,       32;
  4:  73,  1872,  10500,   17600,     9360,     1536,        64;
  5:  99,  3626,  31164,   88200,    90160,    32928,      4032,      128;
  6: 129,  6400,  78400,  338688,   580160,   401408,    107520,    10240,      256;
  7: 163, 10530, 174528, 1072512,  2794176,  3217536,   1612800,   331776,    25344,     512;
  8: 201, 16400, 353700, 2937600, 10866240, 18966528,  15724800,  5990400,   979200,   61440,   1024;
  9: 243, 24442, 665500, 7187400, 35893440, 88792704, 111234816, 69696000, 20908800, 2787840, 146432, 2048;

%F Formula

The row polynomials satisfy (2*n^3+8*n^2+11*n+6+(4*n^2+9*n+2)*x+(2*n+4)*x^2)*P(n,x) = (2*n^3+12*n^2+25*n+18+(12*n^3+52*n^2+65*n+18)*x+(24*n^2+52*n+28)*x^2+(12*n+16)*x^3)*P(n-1,x) + ((4*n^2+14*n+12)*x-(8*n^2+20*n+14)*x^2-(16*n+16)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 9 + 16*x + 4*x^2.

%t Mathematica

P[0, x_] := 9 + 16*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 12*n^2 + 25*n + 18 + (12*n^3 + 52*n^2 + 65*n + 18)*x + (24*n^2 + 52*n + 28)*x^2 + (12*n + 16)*x^3)*P[n - 1, x] + ((4*n^2 + 14*n + 12)*x - (8*n^2 + 20*n + 14)*x^2 - (16*n + 16)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 8*n^2 + 11*n + 6 + (4*n^2 + 9*n + 2)*x + (2*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 16*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3 + 12*n^2 + 25*n + 18 + (12*n^3 + 52*n^2 + 65*n + 18)*x + (24*n^2 + 52*n + 28)*x^2 + (12*n + 16)*x^3)*v[i - 1] + ((4*n^2 + 14*n + 12)*x - (8*n^2 + 20*n + 14)*x^2 - (16*n + 16)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 8*n^2 + 11*n + 6 + (4*n^2 + 9*n + 2)*x + (2*n + 4)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Delannoy paths counted by number of weak peaks.?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023